Understanding Engineering Mathematics

in any event it only requires care to use=and appropriately. In algebra we consider
only exact solutions, always use=and avoid decimals wherever possible.
The simplest equation is of course

x=α

whereαis a given number. Here the solution is trivial and immediate. Writing the equation
in the form

x−α= 0

is only a small disguise, and it is often convenient to write equations in this way, with an
algebraic expression equated to zero. The most general form of such alinear equationin
xis
ax+b= 0

wherea,bare given constants anda =0. The solution is of coursex=−

b

a

, as noted in

Section 2.2.4.
The solution of more complicated polynomial equations, such as quadratics, usually
amounts to trying to pick out such linear factors which can be solved separately, as above.
This is most easily seen with quadratic equations in the case when they can be factorised.
The most generalquadratic equationinxis

ax^2 +bx+c= 0

wherea( = 0 ),b,care given numbers. If the left-hand side factorises then this gives us
two linear equations to solve.

Example

Solve the equation 3x^2 + 5 x− 2 = 0
Factorising, we have

3 x^2 + 5 x− 2 ≡( 3 x− 1 )(x+ 2 )= 0

So the left-hand side can only be zero if

3 x− 1 =0 whencex=^13

or x+ 2 =0 whencex=− 2

Notice the use of a key property of real numbers – ifab=0 then one or both ofa,bis
zero – this property does not hold, for example, for matrices (Chapter 13).
So the two solutions arex=−2,^13.
A quadratic equation must of course always have two solutions, although these may be
equal, or complex (Chapter 12).

Example

Solvex^2 + 4 x+ 4 = 0

x^2 + 4 x+ 4 ≡(x+ 2 )^2 = 0

giving two identical solutionsx=−2.